The Tory leader, who has proclaimed his ‘hope and wish’ that Scots vote to stay in the United Kingdom, made the offer in a speech in Edinburgh where he eulogised the 300-year political union between Scotland and England.

‘This doesn’t have to be the end of the road,’ he said.

‘When the referendum on independence is over, I am open to looking at how the devolved settlement can be improved further.

‘And, yes, that does mean considering what further powers could be devolved.’

The Prime Minister was also in the Scottish capital to hold crunch talks with Scottish first minister Alex Salmond on plans for an independence referendum – a topic the two have clashed on in recent months.

Westminster wants to impose strict restrictions on the terms and timing of any ballot on the issue, but Scottish National Party leader Mr Salmond believes that all the conditions of the vote, including whether 16 and 17-year-olds should be allowed to participate, should be decided by Scotland alone.

The Conservative leader made his case for the Union in the earlier speech in Edinburgh.

Mr Cameron said: ‘I come here today with one simple message: I hope and wish that Scotland will vote to remain part of the United Kingdom.’

Mr Cameron said Scotland had been a ‘pioneering country all its life’.

But he added that the United Kingdom provided a ‘warm and stable home that billions elsewhere envy’.

He argued it was the ‘inspiring model’ of the United Kingdom that could provide the answer to some of the big challenges of these times.

‘I am convinced that both for Scotland, and the United Kingdom, our best days lie ahead of us,’ he added.

In the meeting the prime minister is expected to acknowledge that there are arguments to be made about the future of banking debt and oil reserves, but will insist that these are side issues, and the central point is that the UK will be ‘stronger, safer, richer and fairer’ as a whole.